AI 中文总结
针对发射器、散射体、接收器均运动的标量波动方程逆散射问题，建立时域点相互作用模型，提出结合初始定位与常微分方程组求解的重建算法，数值实验验证其对噪声的鲁棒性。
AI 中文摘要
我们考虑标量波动方程的逆散射问题，其中点发射器、类点散射体和多个接收器均处于运动状态，目标是从散射场的时变测量数据中重建运动类点散射体的轨迹。为此，我们在时域中为运动散射体建立了严格的点相互作用模型，证明了正散射问题的适定性，包括散射场关于散射体轨迹和散射参数的存在性、唯一性及连续依赖性。通过利用延迟时间结构，我们引入了连接发射、散射和观测过程的距离函数，并证明这些函数满足由测量数据确定的耦合非线性常微分方程组。基于该公式，我们提出了一种结合初始定位与推导的常微分方程组求解的重建算法。数值实验表明，该方法对噪声具有准确性和鲁棒性，在中等扰动下重建误差保持稳定。
英文摘要
We consider an inverse scattering problem for the scalar wave equation in which the point emitter, a point-like scatterer, and multiple receivers are all moving. The goal is to reconstruct the trajectory of a moving point-like scatterer from time-dependent measurements of the scattered field. To this end, we establish a rigorous point-interaction model for the moving scatterer in the time domain, proving well-posedness of the forward scattering problem, including existence, uniqueness and continuous dependence of the scattered field on the scatterer trajectory and scattering parameter. By exploiting the retarded-time structure, we introduce distance functions that connect emission, scattering, and observation processes, and show that they satisfy a coupled system of nonlinear ordinary differential equations determined by the measurement data. Based on this formulation, we propose a reconstruction algorithm that combines initial localization with the solution of the derived ODE system. Numerical experiments demonstrate that the method is accurate and robust with respect to noise, with reconstruction errors remaining stable under moderate perturbations.
